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Theorem arwhoma 17384
 Description: An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
arwrcl.a 𝐴 = (Arrow‘𝐶)
arwhoma.h 𝐻 = (Homa𝐶)
Assertion
Ref Expression
arwhoma (𝐹𝐴𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))

Proof of Theorem arwhoma
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 arwrcl.a . . . . . . 7 𝐴 = (Arrow‘𝐶)
2 arwhoma.h . . . . . . 7 𝐻 = (Homa𝐶)
31, 2arwval 17382 . . . . . 6 𝐴 = ran 𝐻
43eleq2i 2843 . . . . 5 (𝐹𝐴𝐹 ran 𝐻)
54biimpi 219 . . . 4 (𝐹𝐴𝐹 ran 𝐻)
6 eqid 2758 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
71arwrcl 17383 . . . . . 6 (𝐹𝐴𝐶 ∈ Cat)
82, 6, 7homaf 17369 . . . . 5 (𝐹𝐴𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
9 ffn 6503 . . . . 5 (𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V) → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)))
10 fnunirn 7010 . . . . 5 (𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)) → (𝐹 ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻𝑧)))
118, 9, 103syl 18 . . . 4 (𝐹𝐴 → (𝐹 ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻𝑧)))
125, 11mpbid 235 . . 3 (𝐹𝐴 → ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻𝑧))
13 fveq2 6663 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
14 df-ov 7159 . . . . . 6 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
1513, 14eqtr4di 2811 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
1615eleq2d 2837 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹 ∈ (𝐻𝑧) ↔ 𝐹 ∈ (𝑥𝐻𝑦)))
1716rexxp 5688 . . 3 (∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻𝑧) ↔ ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦))
1812, 17sylib 221 . 2 (𝐹𝐴 → ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦))
19 id 22 . . . . 5 (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ (𝑥𝐻𝑦))
202homadm 17379 . . . . . 6 (𝐹 ∈ (𝑥𝐻𝑦) → (doma𝐹) = 𝑥)
212homacd 17380 . . . . . 6 (𝐹 ∈ (𝑥𝐻𝑦) → (coda𝐹) = 𝑦)
2220, 21oveq12d 7174 . . . . 5 (𝐹 ∈ (𝑥𝐻𝑦) → ((doma𝐹)𝐻(coda𝐹)) = (𝑥𝐻𝑦))
2319, 22eleqtrrd 2855 . . . 4 (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
2423rexlimivw 3206 . . 3 (∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
2524rexlimivw 3206 . 2 (∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
2618, 25syl 17 1 (𝐹𝐴𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∃wrex 3071  Vcvv 3409  𝒫 cpw 4497  ⟨cop 4531  ∪ cuni 4801   × cxp 5526  ran crn 5529   Fn wfn 6335  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156  Basecbs 16554  domacdoma 17359  codaccoda 17360  Arrowcarw 17361  Homachoma 17362 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-1st 7699  df-2nd 7700  df-doma 17363  df-coda 17364  df-homa 17365  df-arw 17366 This theorem is referenced by:  arwdm  17386  arwcd  17387  arwhom  17390  arwdmcd  17391  coapm  17410
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